**The plastic number**The sequence of side lengths of equilateral triangles in this picture form the

**Padovan sequence** (1,1,1,2,2,3,4,5,7,9...). Just as the Fibonacci sequence is governed by the properties of the

**golden ratio**, the Padovan sequence is governed by the properties of the so-called

**plastic number**.

The

**Padovan sequence** P(n) is sequence is defined by setting P(1)=P(2)=P(3)=1, and then requiring P(n) = P(n–2) + P(n–3) for n > 3. The

**generating function** for the sequence is given by G(x)=(1+x)/(1–x^2–x^3), which means that if this ratio of polynomials is expanded as a power series in x, the coefficient in G(x) of x^n (i.e., x to the nth power) is equal to P(n).

The denominator in the formula for the generating function, 1–x^2–x^3, can be regarded as an algebraic encoding of the recurrence relation P(n)–P(n–2)–P(n–3)=0. An easy calculation involving polynomials shows that the product (1–x^2–x^3)(1–x+x^2) is given by 1–x–x^5. This means that the generating function G(x) can be rewritten so that the denominator polynomial is given by 1–x–x^5, which in turn means that, if n is large enough, the sequence will satisfy the recurrence relation P(n) = P(n–1) + P(n–5).

The picture is an illustration of this last relation. Notice that the big triangle with side 16 is bounded on one side by the

*preceding* triangle in the sequence (of side 12) and the triangle

*five* places earlier in the sequence (of side 4). This corresponds to the fact that P(n) = P(n–1) + P(n–5) for n=12 and P(n)=16.

It turns out that the polynomial 1–x^2–x^3 has exactly one real root, and the

**plastic number** is the reciprocal of this root. Another way to say this is that the plastic number is the unique real solution of the equation x^3=x+1. It is not hard to show using abstract algebra that this solution is an irrational number; the same is true for the golden ratio, which is the larger real solution of the equation x^2=x+1. The decimal expansion of the plastic number is therefore non-recurring; the first few digits are 1.324717957..., and over 10,000,000,000 digits have been computed.

The plastic number is mathematically significant because it is the smallest

**Pisot number**. A Pisot number is a real root of a monic integer polynomial whose other roots are complex numbers of absolute value less than 1. The word

*monic* means that the highest power of x occurring has a coefficient of 1. The connection with the Padovan sequence is that the ratio P(n+1)/P(n), as n becomes large, tends to the plastic number. (In the case of the Fibonacci numbers, the corresponding ratio approaches the golden ratio.)

The Padovan sequence was described by

**Richard Padovan** in a 1994 essay about the Dutch architect

**Hans van der Laan**. Padovan attributed the discovery of the sequence to van der Laan, so

*van der Laan sequence* would have been a more historically accurate name. The reason for the name

*plastic* is too weak to explain convincingly, but it is intended to convey the sense of

*something that can be given a three-dimensional shape*.

The Padovan sequence has several fairly natural interpretations. One of these is that P(n) is the number of ways of writing n+1 as an ordered sum in which each term is either 2 or 3. For example, P(7) is

**4**, and this corresponds to the

**four** ways in which 7+1=8 can be written as an ordered sum of 2s and 3s: 8 = 2+2+2+2 = 3+3+2 = 3+2+3 = 3+3+2.

**Relevant links**Wikipedia on the plastic number:

http://en.wikipedia.org/wiki/Plastic_numberWikipedia on Pisot numbers, also known as

*Pisot–Vijayaraghavan numbers* or

*PV numbers*:

http://en.wikipedia.org/wiki/Pisot–Vijayaraghavan_numberWikipedia on the Padovan sequence, where this picture comes from:

http://en.wikipedia.org/wiki/Padovan_sequenceThe Padovan sequence at the

*On-Line Encyclopedia of Integer Sequences:* http://oeis.org/A000931 (Note that the encyclopedia version contains some additional terms at the beginning relative to the definition of the sequence used here.)

**Irrelevant link**I now have the 1969 song

*Plastic Man* by the Kinks (

http://goo.gl/6dcpbO) stuck in my head. The BBC refused to play the song when it came out because it contains the word “bum”.

(Picture found via Shecky R and Cliff Pickover on Twitter.)

#mathematics #scienceeveryday